Synchronous and asynchronous dynamics in all-to-all coupled networks of identical,
excitatory, current-based, integrate-and-fire (I&F) neurons with delta-impulse coupling currents and
Poisson spike-train external drive are studied. Repeating synchronous total firing events, during
which all the neurons fire simultaneously, are observed using numerical simulations and found to
be the attracting state of the network for a large range of parameters. Mechanisms leading to such
events are then described in two regimes of external drive: superthreshold and subthreshold. In
the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the
oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a
diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed
computationally in networks with random transmission delays. Neuronal voltage probability density
functions (PDFs) and gain curves—graphs depicting the dependence of the network firing rate on
the external drive strength—are analyzed using the steady solutions of the self-consistency problem
for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic
solutions for the gain curves are obtained in several physiologically relevant limits. The absence of
chaotic dynamics is proved for the type of network under investigation by demonstrating convergence
in time of its trajectories.